On the Complexity of Properties of Partial Bijection Semigroups
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909798711164928 |
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| author | Jack, Trevor |
| author_facet | Jack, Trevor |
| contents | We examine the computational complexity of problems in which we are given generators for a partial bijection semigroup and asked to check properties of the generated semigroup. We prove that the following problems are in AC$^0$: (1) enumerating left and right identities and (2) checking if the semigroup is completely regular. We also describe a nondeterministic logspace algorithm for checking if an inverse semigroup given by generators satisfies a fixed semigroup identity that may involve a unary inverse operation. We conclude with an alternative proof that checking membership of a given idempotent in a partial bijection semigroup is a PSPACE-complete problem. The proof reduces from the well-known PSPACE-complete Rectangle Tiling Problem, thereby illustrating a connection between Wang tilings and partial bijection semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_00324 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Complexity of Properties of Partial Bijection Semigroups Jack, Trevor Group Theory We examine the computational complexity of problems in which we are given generators for a partial bijection semigroup and asked to check properties of the generated semigroup. We prove that the following problems are in AC$^0$: (1) enumerating left and right identities and (2) checking if the semigroup is completely regular. We also describe a nondeterministic logspace algorithm for checking if an inverse semigroup given by generators satisfies a fixed semigroup identity that may involve a unary inverse operation. We conclude with an alternative proof that checking membership of a given idempotent in a partial bijection semigroup is a PSPACE-complete problem. The proof reduces from the well-known PSPACE-complete Rectangle Tiling Problem, thereby illustrating a connection between Wang tilings and partial bijection semigroups. |
| title | On the Complexity of Properties of Partial Bijection Semigroups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2101.00324 |